Image Processing Reference
In-Depth Information
Definition 9.4.1 Nearness Relation
LethO,
F
ibe perceptual system and let X, Y⊆O.
A set X is near to a set Y within
the perceptual systemhO,
and there are
A∈O /∼ A , B∈O /∼ B , C∈O /∼ f such that A, B⊆C. If a perceptual system is understood,
then we say briefly that a set X is near to a set Y .
F
i(X
./ F
Y ) iff there areA,B⊆
F
and f∈
F
Definition 9.4.2 Weak Nearness Relation
LethO,
ibe a perceptual system and let X, Y⊆O. A set X is weakly near to a set Y
within the perceptual systemhO,
F
F
i(X./ F Y ) iff there are x∈X and y∈Y and there is
B⊆
F
such that x∼ B y, i.e., set X is weakly near to a set Y .
TABLE 9.1 Relation Symbols
Symbol Interpretation
O Set of perceptual objects,
X X⊆O, set of sample objects,
F A set of probe functions,
B B⊆F,
< reals,
φ φ∈B, where φ i : O−→<, probe function,
B {(x, y)| f (x) = f (y)∀φ∈B},
' B {(x, y)∈O×O|∃φ i ∈B, φ i (x) = φ i (y)},
= B,ε Perceptual tolerance relation,
x /∼ B x /∼ B ={y∈X | y∼ B x}(class),
O /∼ B O /∼ B ={x /∼ B | x∈O}, quotient set,
./ B nearness relation symbol,
./ B weak nearness relation symbol,
./ B,ε
Weak tolerance nearness relation symbol.
Definition 9.4.3 Weak Tolerance Nearness
Let O,
be a perceptual system and let X, Y⊆O,
F
R
. The set X is perceptually near
to the set Y within the perceptual system O,
(X ./ F
F
Y ) iff there exists x∈X, y∈Y
such that x = B y. If a perceptual system is understood, then
we say shortly that a set X is perceptually near to a set Y in a weak tolerance sense of
nearness.
and there is a φ∈
F
,
R
Definition 9.4.4 Tolerance Near Sets
LethO,
ibe a perceptual system and let X⊆O. A set X is a tolerance near set iff there is
Y⊆O such that X ./ F
F
Y . The family of near sets of a perceptual systemhO,
F
iis denoted
by Near F (O).
9.5
Three Tolerance Near Set-based Nearness Measures for
Image Analysis and Comparison
The term tolerance space was introduced by E.C. Zeeman in 1961 in modelling visual per-
ception with tolerances. Tolerance relations are viewed as good models of how one perceives,
how one sees (Zeeman, 1961). Tolerance relations are also considered as a basis for studying
similarities between objects (Peters, 2008c). The basic idea behind using tolerance classes

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